[Matthew wrote](https://forum.azimuthproject.org/discussion/comment/18698/#Comment_18698):

> Some of the objects in this category are \\(\varnothing, \\{\varnothing\\},\\) and \\( \\{\\{\varnothing\\}\\}\\).

That's not the case under my reading of the problem. There are multiple maps \\(\\{\varnothing\\} \to \\{1, 2\\}\\), so \\(\\{\varnothing\\}\\) is not a set with the property we want.

[John wrote](https://forum.azimuthproject.org/discussion/2198/lecture-34-chapter-3-categories):

> There are certain sets S such that there's at most one function from any set to S and also at most one function from S to any set.

I interpreted "any set" in John's problem statement to refer to all sets, not just the class of sets closed under our desired property. If we took the alternative reading, then we get a category containing the null set and all singleton sets; certainly this category is a preorder simply because we demanded it be one, but it is not a partial order for the reason you give.

> Some of the objects in this category are \\(\varnothing, \\{\varnothing\\},\\) and \\( \\{\\{\varnothing\\}\\}\\).

That's not the case under my reading of the problem. There are multiple maps \\(\\{\varnothing\\} \to \\{1, 2\\}\\), so \\(\\{\varnothing\\}\\) is not a set with the property we want.

[John wrote](https://forum.azimuthproject.org/discussion/2198/lecture-34-chapter-3-categories):

> There are certain sets S such that there's at most one function from any set to S and also at most one function from S to any set.

I interpreted "any set" in John's problem statement to refer to all sets, not just the class of sets closed under our desired property. If we took the alternative reading, then we get a category containing the null set and all singleton sets; certainly this category is a preorder simply because we demanded it be one, but it is not a partial order for the reason you give.